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An Introduction to K-Theory An Introduction to K-theory Eric M. Friedlander∗ Department of Mathematics, Northwestern University, Evanston, USA Lectures given at the School on Algebraic K-theory and its Applications Trieste, 14 - 25 May 2007 LNS0823001 ∗[email protected] Contents 0 Introduction 5 1 K ( ), K ( ), and K ( ) 7 0 − 1 − 2 − 1.1 Algebraic K0 of rings . 7 1.2 Topological K0 . 9 1.3 Quasi-projective Varieties . 10 1.4 Algebraic vector bundles . 12 1.5 Examples of Algebraic Vector Bundles . 13 1.6 Picard Group P ic(X) . 14 1.7 K0 of Quasi-projective Varieties . 15 1.8 K1 of rings . 16 1.9 K2 of rings . 17 2 Classifying spaces and higher K-theory 19 2.1 Recollections of homotopy theory . 19 2.2 BG . 20 2.3 Quillen’s plus construction . 22 2.4 Abelian and exact categories . 23 1 2.5 The S− S construction . 24 2.6 Simplicial sets and the Nerve of a Category . 26 2.7 Quillen’s Q-construction . 28 3 Topological K-theory 29 3.1 The Classifying space BU Z . 29 × 3.2 Bott periodicity . 32 3.3 Spectra and Generalized Cohomology Theories . 33 3.4 Skeleta and Postnikov towers . 36 3.5 The Atiyah-Hirzebruch Spectral sequence . 37 3.6 K-theory Operations . 39 3.7 Applications . 41 4 Algebraic K-theory and Algebraic Geometry 42 4.1 Schemes . 42 4.2 Algebraic cycles . 44 4.3 Chow Groups . 46 4.4 Smooth Varieties . 49 4.5 Chern classes and Chern character . 51 4.6 Riemann-Roch . 53 5 Some Difficult Problems 55 5.1 K (Z) . 55 ∗ 5.2 Bass Finiteness Conjecture . 57 5.3 Milnor K-theory . 58 5.4 Negative K-groups . 59 5.5 Algebraic versus topological vector bundles . 60 5.6 K-theory with finite coefficients . 60 5.7 Etale K-theory . 62 5.8 Integral conjectures . 63 5.9 K-theory and Quadratic Forms . 65 6 Beilinson’s vision partially fulfilled 65 6.1 Motivation . 65 6.2 Statement of conjectures . 66 6.3 Status of Conjectures . 67 6.4 The Meaning of the Conjectures . 69 6.5 Etale cohomology . 71 6.6 Voevodsky’s sites . 74 References 75 An Introduction to K-theory 5 0 Introduction These notes are a reasonably faithful transcription of lectures which I gave in Trieste in May 2007. My objective was to provide participants of the Al- gebraic K-theory Summer School an overview of various aspects of algebraic K-theory, with the intention of making these lectures accessible to partici- pants with little or no prior knowledge of the subject. Thus, these lectures were intended to be the most elementary as well as the most general of the six lecture series of our summer school. At the end of each lecture, various references are given. For example, at the end of Lecture 1 the reader will find references to several very good expositions of aspects of algebraic K-theory which present their subject in much more detail than I have given in these lecture notes. One can view these present notes as a “primer” or a “course outline” which offer a guide to formulations, results, and conjectures of algebraic K-theory found in the literature. The primary topic of each of my six lectures is reflected in the title of each lecture: 1. K ( ), K ( ), and K ( ) 0 − 1 − 2 − 2. Classifying spaces and higher K-theory 3. Topological K-theory 4. Algebraic K-theory and Algebraic Geometry 5. Some Difficult Problems 6. Beilinson’s vision partially fulfilled Taken together, these lectures emphasize the connections between alge- braic K-theory and algebraic geometry, saying little about connections with number theory and nothing about connections with non-commutative geom- etry. Such omissions, and many others, can be explained by the twin factors of the ignorance of the lecturer and the constraints imposed by the brevity of these lectures. Perhaps what is somewhat novel, especially in such brief format, is the emphasis on the algebraic K-theory of not necessarily affine schemes. Another attribute of these lectures is the continual reference to topological K-theory and algebraic topology as a source of inspiration and intuition. 6 E.M. Friedlander We very briefly summarize the content of each of these six lectures. Lec- ture 1 introduces low dimensional K-theory, with emphasis on K0(X), the Grothendieck group of finitely generated projective R-modules for a (com- mutative) ring R if Spec R = X, of topological vector vector bundles over X if X is a finite dimensional C.W. complex, and of coherent, locally free -modules if X is a scheme. Without a doubt, a primary goal (if not the OX primary goal) of K-theory is the understanding of K0. The key concept discussed in Lecture 2 is that of “homotopy theo- retic group completion”, an enriched extension of the process introduced by Alexander Grothendieck of taking the group associated to a monoid. We briefly consider three versions of such a group completion, all due to 1 Daniel Quillen: the plus-construction, the S− S-construction, and the Q- construction. In this lecture, we remind the reader of simplicial sets, abelian categories, and the nerve of a category. The early development of topological K-theory by Michael Atiyah and Fritz Hirzebruch has been a guide for many algebraic K-theorists during the past 45 years. Lecture 3 presents some of machinery of topological K-theory (spectra in the sense of algebraic topology, the Atiyah-Hirzebruch spectral sequence, and operations in K-theory) which reappear in more recent devel- opments of algebraic K-theory. In Lecture 4 we discuss the relationship of algebraic K-theory to the study of algebraic cycles on (smooth) quasi-projective varieties. In particu- lar, we remind the reader of the definition of Chow groups of algebraic cycles modulo rational equivalence. The relationship between algebraic K-theory and algebraic cycles was realized by Alexander Grothendieck when he first introduced algebraic K-theory; indeed, algebraic K0 figures in the formula- tion of Grothendieck’s Riemann-Roch theorem. As we recall, one beautiful consequence of this theorem is that the Chern character from K0(X) to CH∗(X) of a smooth, quasi-projective variety X is a rational equivalence. In order to convince the intrigued reader that there remain many funda- mental questions which await solutions, we discuss in Lecture 5 a few difficult open problems. For example, despite very dramatic progress in recent years, we still do not have a complete computation of the algebraic K-theory of the integers Z. This lecture concludes somewhat idiosyncratically with a dis- cussion of integral analogues of famous questions formulated in terms of the “semi-topological K-theory” constructed by Mark Walker and the author. The final lecture could serve as an introduction to Professor Weibel’s lectures on the proof of the Bloch-Kato Conjecture. The thread which orga- An Introduction to K-theory 7 nizes the effort of many mathematicians is a list of 7 conjectures by Alexander Beilinson which proposes to explain to what extent algebraic K-theory pos- sesses properties analogous to those enjoyed by topological K-theory. We briefly discuss the status of these conjectures (all but the Beilinson-Soul´e vanishing conjecture appear to be verified) and discuss briefly the organiza- tional features of the motivic spectral sequence. We conclude this Lecture 6, and thus our series of lectures, with a very cursory discussion of etale cohomology and Grothendieck sites introduced by Vladimir Voevodsky in his dazzling proof of the Milnor Conjecture. 1 K ( ), K ( ), and K ( ) 0 − 1 − 2 − Perhaps the first new concept that arises in the study of K-theory, and one which recurs frequently, is that of the group completion of an abelian monoid. The basic example to keep in mind is that the abelian group of integers Z is the group completion of the monoid N of natural numbers. Recall that an abelian monoid M is a set together with a binary, associative, commutative operation + : M M M and a distinguished element 0 M which serves × → ∈ as an identify (i.e., 0 + m = m for all m M). Then we define the group ∈ completion γ : M M + by setting M + equal to the quotient of the free → abelian group with generators [m], m M modulo the subgroup generated ∈ by elements of the form [m] + [n] [m + n] and define γ : M M + by − → sending m M to [m]. We frequently refer to M + as the Grothendieck ∈ group of M. The group completion map γ : M M + satisfies the following universal → property. For any homomorphism φ : M A from M to a group A, there → exists a unique homomorphism φ+ : M + A such that φ+ γ = φ : M A. → ◦ → 1.1 Algebraic K0 of rings This leads almost immediately to K-theory. Let R be a ring (always assumed associative with unit, but not necessarily commutative). Recall that an (always assumed left) R-module P is said to be projective if there exists another R-module Q such that P Q is a free R-module. ⊕ Definition 1.1. Let (R) denote the abelian monoid (with respect to ) P ⊕ of isomorphism classes of finitely generated projective R-modules. Then we define K (R) to be (R)+. 0 P 8 E.M. Friedlander Warning: The group completion map γ : (R) K (R) is frequently not P → 0 injective. Exercise 1.2. Verify that if j : R S is a ring homomorphism and if P is → a finitely generated projective R-module, then S P is a finitely generated ⊗R projective S-module.
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